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GEOMETRY BASIC GEOMETRY Point: A point is a location in space. It is represented by a dot. Point are usually named with a upper case letter. For example, we refer to the following as "point A" We can also say a point is the intersection of two lines. Line: A line is a collection of points that extend forever. The following is a line. The two arrows are used to show that it extends forever. We can also say that a line is the intersection between two planes. Line segment: A line segment is part of a line. The following is a segment. A segment has two endpoints. The endpoints in the following segments are A and F. Notice also that the line has no endpoints. 09_BASIC GEOMETRY Página 1 de 16 GEOMETRY Ray: A ray is a collection of points that begin at one point (an endpoint) and extend forever on one direction. The following is a ray. Angle: Two rays with the same endpoint is an angle. The following is an angle. Plane: A plane is a flat surface like a piece of paper. It extends in all directions. We can use arrows to show that it extends in all directions forever. The following is a plane RELATIONSHIP BETWEEN TWO STRAIGHT LINES Two lines can be: Parallel lines : When two lines never meet in space or on a plane no matter how long we extend them, we say that they are parallel lines The following lines are parallel. 09_BASIC GEOMETRY Página 2 de 16 GEOMETRY Intersecting lines: When lines meet in space or on a plane, we say that they are intersecting lines . The following are intersecting lines. Perpendicular lines: A line is perpendicular to another if it meets or crosses it at right angles (90°). TYPES OF LINE STRAIGHT LINE or CURVED LINE In geometry when we say line it is a straight line. POLYGONAL LINE A broken line composed of a finite number of line segments
09_BASIC GEOMETRY Página 3 de 16 GEOMETRY Types of angles Type of Angle Acute Angle
Right Angle
Description an angle that is less than 90° an angle that is 90° exactly Obtuse Angle
an angle that is greater than 90° but less than 180° Straight Angle
an angle that is 180° exactly Reflex Angle
an angle that is greater than 180° and less than 360o 09_BASIC GEOMETRY Página 4 de 16 GEOMETRY Pairs of Angles When parallel lines get crossed by another line (which is called aTransversal), you can see that many angles are the same, as in this example: These angles can be made into pairs of angles which have special names. Transversal line Parallel lines Vertical angles: a=d; c=b; e=h; g=f (Ángulos opuestos por el vértice) Corresponding angles: a=e; b=f; c=g; h=d (ángulos correspondientes) c+e =180o ; d+f=180o a+g= 180o; b+ h=180o http://www.youtube.com/watch?v=8eD3wODClh8 09_BASIC GEOMETRY Página 5 de 16 GEOMETRY Complementary Angles Two Angles are Complementary if they add up to 90 degrees (a Right Angle). These two angles (40° and 50°) are Complementary Angles, because they add up to 90°. Notice that together they make a right angle. But the angles don't have to be together. These two are complementary because 27° + 63° = 90° Supplementary Angles Two Angles are Supplementary if they add up to 180 degrees. These two angles (140° and 40°) are Supplementary Angles, because they add up to 180°. Notice that together they make a straight angle. 09_BASIC GEOMETRY Página 6 de 16 GEOMETRY EXERCISES 1. Calculate the value of every complementary angle 2. Plot and find out in every case the complementary angle 3. Match every angle and its complementary one 4. Calculate the value of every suplementary angle 5. Plot and find out in every case the suplementary angle 09_BASIC GEOMETRY Página 7 de 16 GEOMETRY 6. Match every angle and its complementary one 7. AB and CD are parallel lines and EH is a transversal. What is the size of angle EFB? 8. AB and CD are parallel lines and EH is a transversal. What is the size of angle DGH? 09_BASIC GEOMETRY Página 8 de 16 GEOM
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9 de 16 GEOMETRY TO BISECT AN ANGLE 1.Con centro en V trazo un arco de circunferencia que corta a los lados del ángulo en 1 y 2 2. Con centro en 1 y radio 12 trazo un arco de circunferencia 3. Con centro en 2 y radio 21 trazo un arco de circunferencia 4. Uno V con el punto que han determinado ambos arcos. 10. Bisect the angles given below: CIRCUMFERENCES AND CIRCLES Look at these shapes and think about the difference: The first and the last are lines and the second and the third are surfaces How do I call them? CIRCUMFERENCE: es el lugar geométrico de los puntos del plano que equidistan de un punto llamado centro. A esa distancia se le llama radio CIRCLE: Es la región del plano delimitada por una circunferencia y que posee un área definida. 09_BASIC GEOMETRY Página 10 de 16 GEOMETRY PARTS OF A CIRCUMFERENCE 11. Name the parts that are drawn in these circumferences. 12. Plot the following circumferences. 09_BASIC GEOMETRY Página 11 de 16 GEOMETRY Polygon: When yhe polygonal line is closed we´ve got a polygon. Polygons are 2‐dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up). Which is a polygon? The polygons can be ….Concave or Convex A convex polygon has no angles pointing inwards. More precisely, no internal angles can be more than 180°. If there are any internal angles greater than 180° then it is concave. (Think: concave has a "cave" in it) Concave Convex Regular or Irregular If all angles are equal and all sides are equal, then it is regular, otherwise it is irregular Regular 09_BASIC GEOMETRY Irregular Página 12 de 16 GEOMETRY 7) The straight lines AB and CD: A are parallel B are not parallel because the two given consecutive interior angles do not add to 180° C are not parallel because the two given corresponding angles are not equal D are not parallel because the two given alternate angles are not equal 8) AB and CD are What is the size of angle EFB? parallel lines and EH is a transversal. 9) AB and CD are parallel lines and EH is a transversal. What is the size of angle DGH? 09_BASIC GEOMETRY Página 13 de 16 GEOMETRY 10) ST and UV are parallel lines. c and e are: A consecutive interior angles B alternate angles C vertical angles D corresponding angles 11) ST and UV are parallel lines g and f are: A consecutive interior angles B alternate angles C vertical angles D corresponding angles 12) ST and UV are parallel lines d and e are: A consecutive interior angles B alternate angles C vertical angles D corresponding angles 13) ST and UV are parallel lines d and h are: A consecutive interior angles B alternate angles C vertical angles D corresponding angles 14) Two angles are supplementary and one of them is 31° . What is the size of the other angle? 15) Two angles are complementary and one of them is 31° . What is the size of the other angle? 09_BASIC GEOMETRY Página 14 de 16 GEOMETRY 10. Plot the following: A (1,1); B( 4,3); C(3,6); D(9,1); E(7,7) 1. A line passing through point A. 2. Two intersecting lines that intersected in point B 3. A line parallel to the first one 4. A perpendicular line from C to the BC 4. Four different lines passing through point C 5. A ray which vertex is in point D 6. A line segment which one of its endpoints is E 7. An acute angle 8. An obtuse angle 9. A right angle 10. A plane angle 12. Two supplementary angles 13. Two complementary angles 14. Two parallels line and a transversal line and the names of pairs of angles 09_BASIC GEOMETRY Página 15 de 16 GEOMETRY 09_BASIC GEOMETRY Página 16 de 16